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%I #47 Jul 06 2024 19:48:56
%S 0,1,0,3,2,0,6,5,3,0,10,9,7,4,0,15,14,12,9,5,0,21,20,18,15,11,6,0,28,
%T 27,25,22,18,13,7,0,36,35,33,30,26,21,15,8,0,45,44,42,39,35,30,24,17,
%U 9,0,55,54,52,49,45,40,34,27,19,10,0,66,65,63,60,56,51,45,38,30,21,11,0
%N Array T, read by descending antidiagonals: T(n, k) = k*(2*n + k + 1)/2 for n, k >= 0.
%C Triangle S(n,k) = T(k, n-k), read by rows, is given by S(n,k) = A000217(n) - A000217(k) for n >= 0 and 0 <= k <= n. - _Philippe Deléham_, Mar 07 2013 [Edited by _Petros Hadjicostas_, Nov 20 2019]
%H G. C. Greubel, <a href="/A049780/b049780.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,k) = Sum_{j=1..k} (n+j) = k*(2*n + k + 1)/2.
%F S(n,k) = n*(n+1)/2 - k*(k+1)/2 for n >= 0 and 0 <= k <= n - _Philippe Deléham_, Mar 07 2013 [Edited by _Petros Hadjicostas_, Nov 20 2019]
%F From _Stefano Spezia_, Dec 13 2019: (Start)
%F G.f. for T(n,k): y*(1 - x*y)/((1 - x)^2*(1 - y)^3).
%F E.g.f. for T(n,k): (1/2)*exp(x+y)*y*(2 + 2*x + y). (End)
%F G.f. for S(n,k): x*(1 - x^2*y)/((1 - x*y)^2*(1 - x)^3). - _Petros Hadjicostas_, Dec 14 2019
%e From _Petros Hadjicostas_, Nov 20 2019: (Start)
%e Rectangular array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
%e 0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, ...
%e 0, 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, ...
%e 0, 4, 9, 15, 22, 30, 39, 49, 60, 72, 85, ...
%e 0, 5, 11, 18, 26, 35, 45, 56, 68, 81, 95, ...
%e 0, 6, 13, 21, 30, 40, 51, 63, 76, 90, 105, ...
%e ...
%e From _Philippe Deléham_, Mar 07 2013: (Start)
%e Triangle S(n, k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e 0;
%e 1, 0;
%e 3, 2, 0;
%e 6, 5, 3, 0;
%e 10, 9, 7, 4, 0;
%e 15, 14, 12, 9, 5, 0;
%e 21, 20, 18, 15, 11, 6, 0;
%e 28, 27, 25, 22, 18, 13, 7, 0;
%e 36, 35, 33, 30, 26, 21, 15, 8, 0;
%e ... (End)
%p seq(seq( (n-k)*(n+k+1)/2, k=0..n), n=0..15); # _G. C. Greubel_, Dec 12 2019
%t Table[(n-k)*(n+k+1)/2, {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 12 2019 *)
%o (PARI) T(n,k) = k*(2*n+k+1)/2;
%o for(n=0, 15, for(k=0,n, print1(T(k,n-k), ", "))) \\ _G. C. Greubel_, Dec 12 2019
%o (Magma) [(n-k)*(n+k+1)/2: k in [0..n], n in [0..15]]; // _G. C. Greubel_, Dec 12 2019
%o (Sage) [[(n-k)*(n+k+1)/2 for k in (0..n)] for n in (0..15)] # _G. C. Greubel_, Dec 12 2019
%o (GAP) Flat(List([0..15], n-> List([0..n], k-> (n-k)*(n+k+1)/2 ))); # _G. C. Greubel_, Dec 12 2019
%Y Diagonal sums are in A000330. See also A049777 (triangle without the zeros).
%K nonn,tabl
%O 0,4
%A _Clark Kimberling_
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